MOS scale

This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.
The corresponding expert page for this topic is Mathematics of MOS.

A moment of symmetry (MOS or mos^{[note 1]}) scale is a periodic scale with two properties. 1) It has only two step sizes, where size means the size in cents. 2) not just the steps (i.e. all the 2nds) but also all the 3rds, all the 4ths, and so on (i.e. every interval class) occur in only two sizes. However, the octave occurs in only one size.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

Examples

The most widely used MOS scale is the diatonic scale. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).

interval classes in the 5L 2s MOS scale
interval class	small version		large version
interval class	quality	size	quality	size
2nds (1 step)	minor	s	major	L
3rds (2 steps)	minor	1L + 1s	major	2L
4ths (3 steps)	perfect	2L + 1s	augmented	3L
5ths (4 steps)	diminished	2L + 2s	perfect	3L + 1s
6ths (5 steps)	minor	3L + 2s	major	4L + 1s
7ths (6 steps)	minor	4L + 2s	major	5L + 1s
8ves (7 steps)	perfect	5L + 2s	(only one version)

Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

The only other widely used MOS scale is 2L 3s. Among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs).

See the catalog of MOS for other MOS scales.

Periods and generators

Every MOS scale can be generated by stacking a certain interval called the generator and octave-reducing (or more generally, period-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produces a MOS, because there are more than 2 sizes of each interval class.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...

Step ratio spectrum

The step ratio is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The TAMNAMS system has names for specific ratios and also ranges of ratios.

When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

5L 2s step ratios in various edos
example edo	step ratio	TAMNAMS name	likely temperament interpretations
12	2:1	basic	Meantone or Schismatic
19	3:2	soft	Meantone
22	4:1	superhard	Archy or Superpyth

Naming

Every MOS can be uniquely specified by giving its signature, i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either diatonic or anti-diatonic = 2L 5s).

By default, the equave of a MOS is assumed to be 2/1. To specify a non-octave equave, "⟨equave⟩" is placed after the signature, e.g. 4L 5s⟨3/1⟩. Using angle brackets (⟨ and ⟩) is recommended; using greater-than and less-than signs ("<equave>") can also be done, but this can conflict with HTML and other uses of these symbols.

Several naming systems have been proposed for MOSes, which can be seen at MOS naming.

History and terminology

The term MOS, and the method of scale construction it entails, were invented by Erv Wilson in 1975. His original paper is archived on Anaphoria.com here: Moments of Symmetry. There is also an introduction by Kraig Grady here: Introduction to Erv Wilson's Moments of Symmetry.

Sometimes, scales are defined with respect to a period and an additional equivalence interval, the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called Multi-MOSes. MOSes in which the equivalence interval is equal to the period are sometimes called Strict MOSes. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term distributionally even scale, with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as well-formed scales, the term used in the 1989 paper by Norman Carey and David Clampitt^[1]. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music (e.g. A B C E F A), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

Equivalent definitions and generalizations

A scale is a MOS if and only if it satisfies one of the following equivalent criteria:

Maximum variety 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.)
Binary and has a generator: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.
Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.

Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of balance and distributional evenness provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see Mathematics of MOS.

Properties

Basic properties

For every MOS scale with an octave period (which is usually the octave), if x-edo is the collapsed tuning (where the small step vanishes) and y-edo is the equalized tuning (where the large (L) step and small (s) step are the same size), then by definition it is an xL (y − x)s MOS scale, and the basic tuning where L = 2s is thus (x + y)-edo. This is also true if the period is 1\p, that is, 1 step of p-edo, which implies that x and y are divisible by p, though note that in that case (if p > 1) you are considering a "multiperiod" MOS scale.
More generally, whenever px-edo and py-edo are used to define two vals (usually but not necessarily through taking the patent vals) while simultaneously also being used to define the pxL (py − px)s MOS scale (where p is the number of periods per octave), then the px & py temperament corresponds to that MOS scale, and adding x and/or y corresponds to tuning closer to x-edo and/or y-edo respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)
For the mathematically-inclined, we can say that whenever we consider a MOS with X/p notes per period in the collapsed tuning and Y/p notes per period in the equalized tuning and p periods per tempered octave (or more generally tempered equave), and whenever we want to associate that MOS with the X & Y rank 2 temperament*, we can say that any natural-coefficient linear combination of vals ⟨X ...] and ⟨Y ...] (where X < Y) corresponds uniquely to a tuning of the X & Y rank 2 temperament between X-ET and Y-ET (inclusive) iff gcd(a, b) = 1, because if k = gcd(a, b) > 1 then the val a⟨X ...] + b⟨Y ...] has a common factor k in all of its terms, meaning it is guaranteed to be contorted. The tuning corresponding to the rational a/b is technically only unique up to (discarding of) octave stretching (or more generally equave-tempering).

The period of this temperament is 1\gcd(X, Y), and the rational a/b is very closely related to the step ratio of the corresponding MOS scale, because 1⟨X ...] + 0⟨Y ...] is the L = 1, s = 0 tuning while 0⟨X ...] + 1⟨Y ...] is the L = 1, s = 1 tuning and 1⟨X ...] + 1⟨Y ...] is the L = 2, s = 1 tuning, so that L = a + b and s = b and therefore:

1/(step ratio) = s/L = b/(a + b) implying step ratio = (a + b)/b ≥ 1 for natural a and b, where if b = 0 then the step ratio is infinite, corresponding to the collapsed tuning.^{[note 2]}

Every MOS scale has two child MOS scales. The two children of the MOS scale aL bs are (a + b)L as (generated by generators of soft-of-basic aL bs) and aL (a + b)s (generated by generators of hard-of-basic aL bs).
Every MOS scale (with a specified equave Ɛ ), excluding aL as⟨Ɛ ⟩, has a parent MOS. If a > b, the parent of aL bs is bL (a − b)s; if a < b, the parent of aL bs is aL (b − a)s.

Advanced discussion

See:

Mathematics of MOS, a more formal definition and a discussion of the mathematical properties.
- Recursive structure of MOS scales, a description of how MOS scales are recursive and how one scale can be converted into a related scale.
- MOS scale family tree, a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.
Generator ranges of MOS, organized by number of scale steps and quantity of L/s steps.
MOS diagrams, visualizations of the MOS process.
How to Find Linear Temperaments, by Graham Breed

Variations

MODMOS scales are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of L − s, the "chroma".
Muddles are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
MOS cradle is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.
Operations on MOSes

Listen

This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.

Notes

↑ The acronym "MOS" is generally pronounced em-oh-ess, while the anacronym "mos", more common in informal and experimental settings, is generally pronounced moss. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.
↑ It is important to note that the correspondence to the X & Y rank 2 temperament only works in all cases if we allow the temperament to be contorted on its subgroup; alternatively, it works if we exclude cases where X & Y describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the 5-limit (having a generator of a semifourth, corresponding to 5L 14s), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered ~4/3) or we exclude it because of its contortion.

References

↑ Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", Music Theory Spectrum, Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.

[1] The acronym "MOS" is generally pronounced em-oh-ess, while the anacronym "mos", more common in informal and experimental settings, is generally pronounced moss. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.

[3] It is important to note that the correspondence to the X & Y rank 2 temperament only works in all cases if we allow the temperament to be contorted on its subgroup; alternatively, it works if we exclude cases where X & Y describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the 5-limit (having a generator of a semifourth, corresponding to 5L 14s), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered ~4/3) or we exclude it because of its contortion.

[2] Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", Music Theory Spectrum, Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.

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